Project #: DDD2009-13

Mentors: Gene Fiorini, gfiorini at dimacs.rutgers.edu, DIMACS

Extremal graph theory is a branch of graph theory that is interested in studying the following scenario: Suppose G is a family of graphs, P a property and an invariant for each . We wish to determine a value m such that whenever for , then G has property P. Those graphs for which and G does not have property P are said to be the extremal graphs with respect to property P and invariant . For example, suppose G is the family of simple graphs on n vertices, n a positive integer. Moreover, suppose P is the property that "G contains a cycle," and the invariant is the number of edges of G, . Then (the number of edges a graph on n vertices can possess and still not contain a cycle) and the extremal graphs are trees on n vertices.

Consider all k-partite simple graphs (k = 2, 3) on a finite set of vertices. The objective of this project is to explore bounds on the number of edges that k-partite graphs can have and still be of large girth. That is, what is the maximum number of edges a bipartite or tripartite graph can have and avoid cycles of length n (n = 3, 4, 5, ?)?